As part of the EUROCS (EUROpean Cloud Systems study) project, cloud-resolving model (CRM) simulations and parallel single-column model (SCM) tests of the sensitivity of moist atmospheric convection to mid-tropospheric humidity are presented. This sensitivity is broadly supported by observations and some previous model studies, but is still poorly quantified. Mixing between clouds and environment is a key mechanism, central to many of the fundamental differences between convection schemes. Here, we define an idealized quasi-steady 'testbed', in which the large-scale environment is assumed to adjust the local mean profiles on a timescale of one hour. We then test sensitivity to the target profiles at heights above 2 km. Two independent CRMs agree reasonably well in their response to the different background profiles and both show strong deep precipitating convection in the more moist cases, but only shallow convection in the driest case. The CRM results also appear to be numerically robust. All the SCMs, most of which are one-dimensional versions of global climate models (GCMs), show sensitivity to humidity but differ in various ways from the CRMs. Some of the SCMs are improved in the light of these comparisons, with GCM improvements documented elsewhere.